Logarithmic conjugacy of a repeated-root Euler operator ($x^2{D}^2-xD+1=x(d^2/d{t}^2)x^{-1},t=\log x$)

For $x>0$, write $t=\log x$ and $y=xv(t)$. The chain rule gives $y^{\prime }=v+v_t$, $y^{\prime \prime }=x^{-1}(v_t+v_{tt})$, and thereforeOn $1<x<e^{\ell }$ with homogeneous Dirichlet boundary conditions, the eigenfunctions are $x\sin (n\pi \log x/\ell )$ and the eigenvalues are $-(n\pi /\ell {)}^2$. The weighted inner product with weight $x^{-3}$ becomes ${\int }_0^{\ell }\overline{v}wdt$, so their squared norms are $\ell /2$ and their orthogonality is exactly that of the Fourier sine series. The weight is the factor that makes this conjugacy preserve the inner product.